We assume here that the background turbulence is of Kolmogorov type with
constant energy flux over the spectrum,
i.e., the velocity fluctuations spectrum for the incompressible part
of turbulence in the
range of wave numbers \(k_0<k<k_\nu \)  is
\(E(k) = - d \bar{\tau }(k) / dk\) , where the function \(\bar{\tau }(k) =(k / k_{0})^{1-q}\)  with \(1 < q < 3\)  being the
exponent of the turbulent kinetic energy spectrum.
The condition \(q>1\)  corresponds to finite turbulent kinetic energy
for very large fluid Reynolds numbers, while \(q<3\)  corresponds to finite
dissipation of the turbulent kinetic energy at the viscous scale
[1]}, [2]}, [3]}, [4]}.
Similarly, the turbulent kinetic energy spectrum
for the compressible part
of turbulence is \(E_c(k) = - d \bar{\tau }_c(k) / dk\) , where the function \(\bar{\tau }_c(k) =(k / k_{0})^{1-q_c}\)  with \(1 < q_c < 3\) .
For instance, the exponent of the incompressible part of
the turbulent kinetic energy spectrum
\(q=5/3\)  (the Kolmogorov spectrum), while the exponent of the compressible part of
the spectrum \(q_c=2\)  (for the Burgers turbulence with shock waves).
The correlation time for a compressible turbulence in the \({k}\)  space is [5]}
\(\tau (k) = {2\tau _0 \over 1+\sigma _c} \, \Big [\bar{\tau }(k) + \sigma _c \,\bar{\tau }_c(k)\Big ] .\) 
